Mechanical modulation of spontaneous emission of nearby nanostructured black phosphorus

HongQian Mu; TongBiao Wang; DeJian Zhang; WenXing Liu; TianBao Yu; QingHua Liao

doi:10.1364/OE.414380

1. Introduction

Before the pioneering work of Purcell in 1946, spontaneous emission has been considered the intrinsic radiation properties of atoms or molecules [1]. The spontaneous emission decay rate can be strongly modified when a quantum emitter is placed near solid structures rather than in vacuum [1]. The possibility of tailoring light–matter interactions has been an interesting topic in optics. A large number of artificial materials and structures have been proposed to modulate the spontaneous emission of atoms, molecules, and fluorescent materials [2,3]. In particular, the progress on nanofabrication techniques is due to the use of nanometric structures, such as nanocavities [2], antennas [4,5], tips [6–8], and metamaterials [9–15]. These structures are used to modify the spontaneous emission of quantum emitter and fluorescence of single molecules. Very recently, the spontaneous emission in anisotropic dielectrics [16] and the optical features of potential sigle-photon sources in two-dimensional silicon carbide monolayers [17] have also been studied. At present, light–matter interactions at a quantum level can be controlled due to the development of plasmonics [18]. When a quantum emitter is placed nearby a plasmonic structure, a strong enhancement of the local field can be achieved [19–21]. Vladimirova et al. studied the modification of the resonance fluorescence spectrum of the two-level atom in close proximity to a metal sphere, they demonstrated that the radius of nanosphere, polarization of the incident radiation, and the position of atom can affect the local field enhancement and the modification of the total decay rate of the atom nearby the nanosphere [20]. Akselrod et al. showed that directional emission, high radiative quantum efficiency, and a large spontaneous emission rate can be simultaneously achieved by utilizing a film-coupled metal nanocube system with emitters embedded in the dielectric gap region [21]. The emerging two-dimensional (2D) material graphene with extraordinary electronic and optical properties [22–24] is a promising plasmonic material. Graphene can support extremely confined surface plasmons due to its high tunability by gate voltage; thus, it can be used as a platform to enhance light–matter interactions [25]. Graphene-based structures have been extensively used to control the spontaneous emission in recent years [26–30].

In addition to graphene, monolayer black phosphorus (BP) is intensively investigated recently due to its unique characteristic including direct bandgap [31], excellent carrier mobility [32,33], strong light–matter interactions in the mid-infrared frequency range [34]. In contrast to the in-plane isotropic property of graphene, BP exhibits an anisotropic optical and electronic nature because its crystal structure is a repeating puckered honeycomb along the armchair (AC) direction. The optical conductivity of BP along the AC direction is different from that along the zigzag (ZZ) direction. BP can also support plasmons, but the natural in-plane anisotropy of BP result in the plasmon dispersion dependent on the direction. Because of its novel electronic and optical properties [32,35–40], BP can be utilized to develop novel polarization-dependent optoelectronic devices, such as optical modulators [41,42], tunable polarization rotators [40,43], polarization-sensitive photodetectors [44], and plasmonic-assisted photovoltaic devices [45]. It also provides the possibilities of devising BP-based plasmons in the applications of optical electric detection and sensing at mid-infrared region [46].

In this study, we investigate the spontaneous emission of a quantum emitter perpendicular and parallel to the BP sheet. For the quantum emitter parallel to the BP sheet, the spontaneous emission decay rate can be modulated by mechanical rotation of BP sheet. The spontaneous emission of quantum emitter perpendicular and parallel to BP ribbon arrays (BPRAs) is studied in detail. Types I and II of BPRAs can be obtained by tailoring the BP sheet along the AC and ZZ directions, respectively. The Purcell factor of quantum emitter is dependent on the electron doping, ribbon width, and the rotation angle of BPRAs. When the distance between the quantum emitter and type I is approximately 1/1000 wavelength, the Purcell factor either decreases or increases with the increase in rotation angle. This condition is closely related to the electron doping of BP. For the quantum emitter parallel to type II, the Purcell factor decreases rapidly compared with that for quantum emitter parallel to type I. The Purcell factor decreases with the increase in rotation angle increases regardless of electron doping. The findings in this study provide a new method to modulate the spontaneous emission of quantum emitter.

2. Theoretical method

The schematic of the spontaneous emission of a quantum emitter nearby the BP sheet placed in the xy plane is depicted in Fig. 1. Owing to its deep subwavelength size, the quantum emitter can be modeled as an electric dipole. The configurations of the dipole moment perpendicular and parallel to the xy plane are displayed in Figs. 1(a) and 1(b), respectively. The spontaneous emission decay rate of a quantum emitter in proximity to a surface can be expressed by [47]

(1)$$\Gamma = \frac{{2k_0^2|\vec{p}{|^2}}}{{\hbar {\varepsilon _0}}}\{{{{\vec{u}}_p}{\mathop{\rm Im}\nolimits} [G(\vec{r},\vec{r},\omega )]{{\vec{u}}_p}} \}, $$

where k=ω/c is the wave vector in vacuum, ω is the angular frequency of electromagnetic waves, and c is the speed of light in vacuum. ħ is the reduced Planck constant, and ε₀ is the dielectric constant in vacuum. $\vec{p}$ is the dipole moment of the quantum emitter, and $\vec{u}{}_p$ is the unitary vector in the direction of the dipole moment. $G(\vec{r},\vec{r},\omega )$ is the dyadic Green’s function of the system and can be expressed as the following formula for the 2D anisotropic surface.

(2)$$G(\vec{r},\vec{r},\omega ) = \frac{i}{{8{\pi ^2}}}\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {({R_{ss}}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over M} }_{ss}} + {R_{sp}}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over M} }_{sp}} + {R_{ps}}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over M} }_{ps}} + {R_{pp}}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over M} }_{pp}}){e^{2i{k_z}z}}d{k_x}d{k_y}} } , $$

where ${R_{ij}}$(i, j = s, p) is the matrix elements of the tensor reflection coefficient $R$ related to incident s- and p-polarized waves. k_x, k_y, and k_z are the components of wave vector along x, y, and z directions, respectively. Matrix ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over M} _{ij}}$(i, j = s, p) can be expressed as

(3)$$\begin{array}{{c}}{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over M} _{ss}} = \frac{1}{{{k_z}k_\rho ^2}}\left( {\begin{array}{{@{}ccc@{}}} {k_y^2}&{ - {k_x}{k_y}}&0\\ { - {k_x}{k_y}}&{k_x^2}&0\\ 0&0&0 \end{array}} \right), {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over M} _{sp}} = \frac{1}{{{k_0}k_\rho ^2}}\left( {\begin{array}{{@{}ccc@{}}} { - {k_x}{k_y}}&{ - k_y^2}&{ - {k_y}k_\rho^2/{k_z}}\\ {k_x^2}&{{k_x}{k_y}}&{{k_x}k_\rho^2/{k_z}}\\ 0&0&0 \end{array}} \right),\\ {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over M} _{ps}} = \frac{1}{{{k_0}k_\rho ^2}}\left( {\begin{array}{{@{}ccc@{}}} {{k_x}{k_y}}&{ - k_x^2}&0\\ {k_y^2}&{ - {k_x}{k_y}}&0\\ { - {k_y}k_\rho^2/{k_z}}&{{k_x}k_\rho^2/{k_z}}&0 \end{array}} \right), {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over M} _{pp}} = \frac{{{k_z}}}{{{k_0^2}k_\rho ^2}}\left( {\begin{array}{{@{}ccc@{}}} { - k_x^2}&{ - {k_x}{k_y}}&{ - {k_y}k_\rho^2/{k_z}}\\ { - {k_x}{k_y}}&{ - k_y^2}&{ - {k_y}k_\rho^2/{k_z}}\\ {{k_x}k_\rho^2/{k_z}}&{{k_y}k_\rho^2/{k_z}}&{k_\rho^4/{k_z^2}} \end{array}} \right),\end{array}$$

where ${k_0} = \sqrt {k_x^2 + k_y^2 + k_z^2} $, and ${k_\rho } = \sqrt {k_x^2 + k_y^2} $ is the wave vector parallel to the xy plane. Reflection matrix elements R_ij must be obtained to determine the spontaneous emission decay rate denoted in Eq. (1). For the 2D anisotropic BP sheet, the AC and ZZ directions of BP are assumed along x- and y-axes, respectively. Its optical conductivity can be described by a tensor with zero off-diagonal conductivity elements.

(4)$$\hat{\sigma } = \left( {\begin{array}{{cc}} {{\sigma_{xx}}}&0\\ 0&{{\sigma_{yy}}} \end{array}} \right). $$

Fig. 1. (a) Schematic of spontaneous emission of a quantum emitter perpendicular to the BP sheet. (b) Schematic of spontaneous emission of an emitter parallel to the BP sheet. (c) BP sheet placed in the Cartesian coordinates with AC and ZZ directions are along x- and y-axes, respectively. (d) Rotated BP sheet with an angle θ with respect to x-axis.

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Monolayer BP exhibits high anisotropy of in-plane electron effective mass and conductivity. The optical conductivity of monolayer BP can be approximated by the Drude model [48]. In the mid-infrared range, the conductivity components of monolayer BP can be expressed as [39,40]

(5)$${\sigma _{jj}}(\omega ) = \frac{{i{D_j}}}{{\pi (\omega + {{i\eta } / \hbar })}},\;\; {D_j} = \pi {e^2}\frac{n}{{{m_j}}}$$

where D_j (j = x, y) is the Drude weight, η is the relaxation rate, n is electron doping, and m_j is the effective mass of electron and can be expressed as

(6)$${m_x} = \frac{{{\hbar ^2}}}{{2{{{\gamma ^2}} / {\Delta + {\eta _c}}}}},\;\; {m_y} = \frac{{{\hbar ^2}}}{{2{\nu _c}}}.$$

The parameters used here are γ = 4a/π eVm, Δ=2 eV, η_c= ħ²/0.4m₀, v_c= ħ²/1.4m₀, and η = 10 meV. $a \approx 0.223$ nm is the scale length of BP, and m₀=9.1×10⁻³¹ kg is the electron mass. Given that BP is an anisotropic material, the reflection characteristic can be denoted by a reflection matrix with matrix elements expressed as [49]

(7)$${R_{ss}} = \frac{{ - {\eta _0}{{\sigma ^{\prime}}_{yy}}(2{Z^p} + {\eta _0}{{\sigma ^{\prime}}_{xx}}) + {\eta _0}^2{{\sigma ^{\prime}}_{xy}}{{\sigma ^{\prime}}_{yx}}}}{{(2{Z^s} + {\eta _0}{{\sigma ^{\prime}}_{yy}})(2{Z^p} + {\eta _0}{{\sigma ^{\prime}}_{xx}}) - {\eta _0}^2{{\sigma ^{\prime}}_{xy}}{{\sigma ^{\prime}}_{yx}}}}, $$

(8)$${R_{sp}} = \frac{{ - 2{c^p}{Z^p}{\eta _0}{{\sigma ^{\prime}}_{xy}}}}{{(2{Z^s} + {\eta _0}{{\sigma ^{\prime}}_{yy}})(2{Z^p} + {\eta _0}{{\sigma ^{\prime}}_{xx}}) - {\eta _0}^2{{\sigma ^{\prime}}_{xy}}{{\sigma ^{\prime}}_{yx}}}}, $$

(9)$${R_{ps}} = \frac{{2{Z^s}{\eta _0}{{{{\sigma ^{\prime}}_{yx}}} / {{c^p}}}}}{{(2{Z^s} + {\eta _0}{{\sigma ^{\prime}}_{yy}})(2{Z^p} + {\eta _0}{{\sigma ^{\prime}}_{xx}}) - {\eta _0}^2{{\sigma ^{\prime}}_{xy}}{{\sigma ^{\prime}}_{yx}}}}, $$

(10)$${R_{pp}} = \frac{{{\eta _0}{{\sigma ^{\prime}}_{xx}}(2{Z^s} + {\eta _0}{{\sigma ^{\prime}}_{yy}}) + {\eta _0}^2{{\sigma ^{\prime}}_{xy}}{{\sigma ^{\prime}}_{yx}}}}{{(2{Z^s} + {\eta _0}{{\sigma ^{\prime}}_{yy}})(2{Z^p} + {\eta _0}{\sigma ^{\prime}}_{xx}) - {\eta _0}^2{{\sigma ^{\prime}}_{xy}}{{\sigma ^{\prime}}_{yx}}}}, $$

where ${\eta _0} = \sqrt {{{{\mu _0}} / {{\varepsilon _0}}}} $ is the impedance in vacuum, ${Z_s} = {{{k_z}} / {{k_0}}}$, ${Z_p} = {{{k_0}} / {{k_z}}}$, and ${c_p} = {{{k_z}} / {{k_0}}}$. $\hat{\sigma ^{\prime}}$ denotes the rotated conductivity tensor.

(11)$$\hat{\sigma }^{\prime} = \left( {\begin{array}{{cc}} {{{\sigma^{\prime}}_{xx}}}&{{{\sigma^{\prime}}_{xy}}}\\ {{{\sigma^{\prime}}_{yx}}}&{{{\sigma^{\prime}}_{yy}}} \end{array}} \right) = {\hat{R}^T}\hat{\sigma }\hat{R}, $$

where $\hat{\sigma }$ is defined in Eq. (1), and $\hat{R}$ defined in the following

(12)$$\hat{R} = \frac{1}{{{k_\rho }}}\left( {\begin{array}{{cc}} {{k_x}}&{ - {k_y}}\\ {{k_y}}&{{k_x}} \end{array}} \right),\;\; {\hat{R}^T} = \frac{1}{{{k_\rho }}}\left( {\begin{array}{{cc}} {{k_x}}&{{k_y}}\\ { - {k_y}}&{{k_x}} \end{array}} \right)$$

are standard rotation matrixes.

3. Spontaneous emission of a quantum emitter nearby BP sheet

The directions of dipole moment vertical and parallel to the BP sheet are considered in the following theoretical calculations. This condition is used to investigate the anisotropic characteristic of BP sheet on spontaneous emission. For the dipole moment vertical to BP sheet, the anisotropic of BP has no effect on the spontaneous emission. For the dipole moment parallel to BP sheet, the rotation of BP sheet may result in the variation of spontaneous emission. In Fig. 1(b), the dipole moment is fixed along the x-axis, and the BP sheet can counterclockwise rotate an angle of θ with respect to x-axis. The spontaneous emission decay rate of a quantum emitter located at a distance z₀ near the surface of homogeneous anisotropic material is normalized to that in free space. Within the electric dipole approximation and weak coupling regime, the normalized energy loss (Purcell factors) of the radiation dipole can be expressed in terms of ${\Gamma / {{\Gamma _0}}}$ [47,50,51]. Here, ${\Gamma _0} = {|{\vec{p}} |^2}{\omega ^3}/3\pi {\varepsilon _0}\hbar {c^3}$ is the spontaneous emission rate in free space. In the following calculations, we use the Purcell factor of the quantum emitter perpendicular and parallel to the xy plane for describing the spontaneous emission.

The dependence of Purcell factor for the quantum emitter perpendicular and parallel to the xy plane at distance z₀ is displayed in Fig. 2(a). The transition frequency of quantum emitter is ν=2.4 THz. For the quantum emitter placed vertically on the top of BP sheet, the Purcell factor does not change when the BP sheet is rotated by a certain angle. ${\Gamma _ \bot }/{\Gamma _0}$ as a function of distance is plotted in black solid line. The Purcell factor decreases when distance z₀ between the BP sheet and quantum emitter increases. When distance z₀ is larger than 10⁻¹ λ₀, the Purcell factor tends to 1.0, as shown in black dotted line in Fig. 2(a), which indicates that the spontaneous emission rate of the quantum emitter gradually approaches that in free space as the distance increases. This condition indicates that the radiation from quantum emitter is emitted to the free space, thereby preventing the excitation of BP plasmon. When distance z₀ is smaller than 10⁻¹ λ₀, the Purcell factor for the quantum emitter perpendicular to BP plane is larger than those parallel to BP plane. We concentrate on the spontaneous emission of a quantum emitter parallel to the BP sheet for different rotation angles θ. The dipole moment is consistent with the AC direction of BP and is along the ZZ direction after BP sheet is rotated to 90°. When the BP sheet rotates at an angle θ with respect to x-axis (the direction of dipole), as shown in Fig. 1(d), the spontaneous emission varies with the change in rotation angle due to the anisotropic characteristic of BP sheet. Purcell factor ${\Gamma _{/{/}}}/{\Gamma _0}$ in parallel direction for different rotation angles of 0°, 30°, 60°, and 90° degree is plotted in Fig. 2(a). The values of ${\Gamma _{/{/}}}/{\Gamma _0}$ are less than that of ${\Gamma _ \bot }/{\Gamma _0}$ for all rotation angles. The value of ${\Gamma _{/{/}}}/{\Gamma _0}$ increases with the increase in rotation angle when distance z₀ is near 10⁻³ λ₀. ${\Gamma _{/{/}}}/{\Gamma _0}$ decreases with the increase in rotation angle when distance z₀ ranges from 10⁻² λ₀ to 10⁻¹ λ₀. The dependence of Purcell factor on rotation angle θ for different electron doping is shown in Fig. 2(b). The distance z₀ between the quantum emitter and BP sheet is fixed at 10⁻¹ λ₀. When the electron doping is small ($n = 0.5 \times {10^{13}}$ cm⁻²), the Purcell factor decreases with the increase in rotation angle. For large electron doping ($n = 1.0 \times {10^{13}}$ cm⁻² and $n = 2.0 \times {10^{13}}$ cm⁻²), the Purcell factor increases with the increase in rotation angle. Considering the quantum emitter is placed parallel to the AC direction ($\theta = {0^\circ }$), the difference between the Purcell factors of quantum emitters nearby BP sheets with low and high electron doping is large. For the quantum emitter placed parallel to the ZZ direction ($\theta = {90^\circ }$), such a difference is extremely small. As has been studied in Ref. [40], the plasmonic property of BP is determined by the imaginary part of optical conductivity which depends on the electron doping. The imaginary part of optical conductivity of BP along AC direction is larger than that along ZZ direction under fixed electron doping and frequency [52,53], that is, ${\mathop{\rm Im}\nolimits} [{\sigma _{AC}}(\omega )] > {\mathop{\rm Im}\nolimits} [{\sigma _{ZZ}}(\omega )]$. Thus, BP shows different plasmonic-like responses to the quantum emitter in different directions. When the electron doping increases, the imaginary part of optical conductivity along both directions will increase. A large Purcell factor can be obtained if the radiation emitted by quantum emitter can excite the BP plasmon controlled by electron doping, which means the electron doping must take a certain value rather than an arbitrary value. Because the direction of quantum emitter is fixed along x-axis, it will experience different optical conductivity when BP rotates. For the electron doping n=0.5×10¹³ cm⁻², the fact that Purcell factor decreases as the AC direction rotates from x to y-axis means a suitable optical conductivity ${\sigma _x}$ can lead to the large Purcell factor. While for electron doping is n=1.0 ×10¹³ cm⁻² or n=2.0×10¹³ cm⁻², the optical conductivity is so large that the radiation emitted by quantum emitter cannot excite BP plasmons efficiently, the Purcell factor decreases obviously when AC direction is along x-axis ($\theta = {0^ \circ }$). When the BP rotates, optical conductivity becomes smaller to approach ${\sigma _x}$, so the Purcell factor increases as BP rotates. We can see that for the BP sheet with electron doping $n = 2.0 \times {10^{13}}$ cm⁻², the Purcell factor of quantum emitter is approximately twice larger for $\theta = {90^ \circ }$ than that for $\theta = {0^ \circ }$. Therefore, the spontaneous emission of quantum emitter can be easily tuned by mechanically rotating the anisotropic BP sheet.

Fig. 2. (a) Purcell factor of a quantum emitter parallel to the BP sheet as a function of distance z₀ for different rotation angles. Black solid line shows the Purcell factor of quantum emitter perpendicular to BP sheet for comparison. The electron doping of BP is fixed at $2.0 \times {10^{13}}$ cm⁻². The black dotted line denotes the valued of Purcell factor in vacuum. (b) Purcell factor of quantum emitter parallel to BP sheet as a function of rotation angle θ for different electron doping with a fixed distance ${z_0} = {10^{ - 3}}{\lambda _0}$. The transition frequency is ν=2.4 THz in all the calculations.

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4. Spontaneous emission of a quantum emitter nearby BPRAs

Two different BPRAs can be obtained by tailoring the BP sheet along the AC and ZZ directions, as shown in Figs. 3(a) and 3(b), respectively. For convenience, the BPRAs tailored along the AC and ZZ directions are denoted as types I and II, respectively. The optical conductivity of BP ribbon arrays can be described by the effective medium theory (EMT) [54]. According to EMT, the components of effective conductivity tensor of type I is expressed as

(13)$${\sigma ^{\prime}_{xx}} = f{\sigma _{AC}}, $$

(14)$${\sigma ^{\prime}_{yy}} = \frac{{{\sigma _{\textrm{ZZ}}}{\sigma _C}}}{{f{\sigma _C} + (1 - f){\sigma _{ZZ}}}}. $$

Fig. 3. (a) BPRAs of type I tailored from BP sheet along the AC direction. (b) BPRAs of type II tailored from BP sheet along the ZZ direction.

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The components of effective conductivity tensor of type II is expressed as

(15)$${\sigma ^{\prime}_{xx}} = \frac{{{\sigma _{\textrm{AC}}}{\sigma _C}}}{{f{\sigma _C} + (1 - f){\sigma _{AC}}}}, $$

(16)$${\sigma ^{\prime}_{yy}} = f{\sigma _{ZZ}}, $$

where ${\sigma _C} ={-} i[\omega {\varepsilon _0}({\varepsilon _1} + {\varepsilon _2})L/2\pi ]\ln \{ \csc [\pi (L - W)/2L]\} $ is an equivalent conductivity associated with the near-field coupling between neighbor BP ribbons. ɛ₁ and ɛ₂ are the relative permittivities of the media above and below the BP ribbons. In this work, the values of ɛ₁ and ɛ₂ are set to 1.0. L is the period of BPRAs, $W$ is the width of the ribbon, and $f = W/L$ is the ratio of BP ribbons. EMT is only valid when the radiation wavelength of quantum emitter is larger than the period of BPRAs [55,56]. Thus, the period of BPRAs is set as $L = 20$ nm.

After replacing the optical conductivities in Eq. (4) with the effective conductivities in Eqs. (13)–(16), the spontaneous emission decay rate of a quantum emitter near BPRAs can be calculated. The Purcell factors of quantum emitters nearby different types of BPRAs are shown in Fig. 4. First, we concentrate on the Purcell factor of a quantum emitter perpendicular to the xy plane as a function of distance z₀ for different electron dopings, as shown in Figs. 4(a) and 4(b). The transition frequency of the quantum emitter is ν=2.4 THz, and the width of BP ribbon is 16 nm. In this situation, the Purcell factors of quantum emitters perpendicular to BPRAs for two different types mostly have no difference. The Purcell factor decreases with the increase in electron doping when distance z₀ is near 10⁻³ λ₀. However, when distance z₀ surpasses a certain value, the Purcell factor increases with the increase in electron doping at a fixed distance z₀. Such a value is smaller in type II than that in type I. When the distance continues to increase to the value of 10⁻¹ λ₀, the Purcell factor is mostly the same with that in vacuum. This condition indicates that means BPRAs have no effect on the spontaneous emission at such a distance. Second, we consider the case of a quantum emitter parallel to BPRAs. The value of Purcell factor for type I is similar with those of quantum emitter vertical to BPRAs. The Purcell factor for type II is approximately two orders of magnitude smaller than that for type I, as clearly shown in Figs. 4(c) and 4(d). When the quantum emitter is parallel to type II, the Purcell factor decreases rapidly to that in vacuum at a distance of 10⁻² λ₀.

Fig. 4. Purcell factors of quantum emitters with transition frequency ν=2.4 THz near BPRAs as a function of distance z₀ for different electron dopings. ${\Gamma _ \bot }/{\Gamma _0}$ and ${\Gamma _{/{/}}}/{\Gamma _0}$ are displayed in top and bottom panels, respectively.

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In Fig. 5, we show the Purcell factor as a function of distance z₀ for different rotation angles when the quantum emitter is parallel to different BPRAs. The cases of types I and II are shown in Figs. 5(a) and 5(b), respectively. The electron doping is set to $2.0 \times {10^{13}}$ cm⁻², and the ribbon width is fixed at 16 nm. For the quantum emitter parallel to type I, the Purcell factor increases with the increase in rotation angle θ when distance z₀ is near 10⁻³ λ₀. However, the Purcell factor decreases with the increase in rotation angle for a fixed distance when distance z₀ is near 10⁻² λ₀, as shown in Fig. 5(a). For the quantum emitter parallel to type II as displayed in Fig. 5(b), the spontaneous emission decay rate decreases significantly compared with that for the quantum emitter parallel to type I. The Purcell factor decreases to the value of that in vacuum when the distance between quantum emitter and type II reaches 10⁻² λ₀. The Purcell factor decreases with the increase in rotation angle θ when distance z₀ ranges from 10⁻³ λ₀ to 10⁻² λ₀. This condition is different from the distance-dependent characteristic between the Purcell factor and rotation angle in type I.

Fig. 5. (a) Purcell factor of the quantum emitter parallel to type I as a function of distance z₀ for different rotation angles. (b) Purcell factor of the quantum emitter parallel to type II as a function of distance z₀ for different rotation angles. The transition frequency of quantum emitter is ν=2.4 THz. The electron doping and ribbon width are fixed at $2.0 \times {10^{13}}$ cm⁻² and 16 nm, respectively.

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As stated in Ref. [53], the ribbon width often plays an important role to the optical characteristic of BPRAs. Thus, we investigate the effect of ribbon width on spontaneous emission. In Fig. 6, the Purcell factors of quantum emitters parallel to types I and II are displayed in left and right panels, respectively. The electron dopings of BP are set to $n = 0.5 \times {10^{13}}$, $n = 1.0 \times {10^{13}}$, and $n = 2.0 \times {10^{13}}$ cm⁻² from top to bottom panels to compare the results with those in Fig. 2(b). The transition frequency of quantum emitter is ν=2.4 THz, and distance z₀ is 10⁻³ λ₀. Three different ribbon widths of $W = 14$, $W = 16$, and $W = 18$ nm are shown in black solid, red dashed, and blue dash-dotted lines, respectively. The Purcell factor enhances obviously for type I than that for BP sheet for all the considered electron dopings. However, the Purcell factor of quantum emitter nearby type II has a significant decrease compared with that nearby BP sheet. Under the same condition, type I plays a more important role to enhance the spontaneous emission than type II. The dependence of Purcell factor on rotation angle for type I is different from that for type II. The Purcell factor of quantum emitter in type I decreases with the increase in rotation angle when the electron doping is low ($n = 0.5 \times {10^{13}}$ cm⁻²). It increases with the increase in rotation angle when the electron doping is high ($n = 2.0 \times {10^{13}}$ cm⁻²). This condition is extremely similar with that of quantum emitter nearby BP sheet. However, when the electron doping is $n = 1.0 \times {10^{13}}$ cm⁻², the relationship between Purcell factor and rotation angle is dependent on the ribbon width. For wide ribbon ($W = 18$ nm), the Purcell factor increases with the increase in rotation angle, which is similar with that in BP sheet with the same electron doping. For narrow ribbon ($W = 14$ nm), the Purcell factor decreases with the increase in rotation angle. For the quantum emitter nearby type II, the Purcell factor decreases with the increase in rotation angle for all three studied electron dopings. Another obvious difference is found between the spontaneous emissions in types I and II. For type I, the Purcell factor increases with the increase in ribbon width at fixed rotation angle θ, except the cases with low electron doping ($n = 0.5 \times {10^{13}}$ cm⁻²) when the rotation angle is larger than 60°. For type II, the Purcell factor decreases with the decrease in ribbon width at the fixed rotation angle.

Fig. 6. Dependence of Purcell factor of quantum emitters parallel to type I (left panel) and type II (right panel) on rotation angle for different ribbon widths. Electron dopings are $n = 0.5 \times {10^{13}}$, $n = 1.0 \times {10^{13}}$, and $n = 2.0 \times {10^{13}}$ cm⁻² from top to bottom panels. The transition frequency of quantum emitter is ν=2.4 THz, and distance z₀ is set to 10⁻³ λ₀.

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5. Conclusions

We studied the spontaneous emission of a quantum emitter on the top of the BP sheet. The spontaneous emission can be modulated mechanically when the quantum emitter is parallel to the BP sheet. For the lower electron doping of BP, the Purcell factor decreases with the increase in rotation angle with respect to quantum emitter, under higher electron doping, the Purcell factor increases with the increase in rotation angle. The spontaneous emissions of quantum emitter nearby two different types of BPRAs are investigated in detail. The BPRAs tailored along the AC direction can enhance the spontaneous emission more significantly than the BPRAs tailored along the ZZ direction. The ribbon width and rotation angle play important role to the spontaneous emission. The Purcell factor can increase with the decreases in ribbon width for the fixed rotation angle in type I. The Purcell factor decreases with the decrease in ribbon width at a fixed rotation angle in type II. The results obtained in this study offer a new way to mechanically modulate the spontaneous emission.

Funding

National Natural Science Foundation of China (11704175, 12064025).

Disclosures

The authors declare no conflicts of interest.

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Mechanical modulation of spontaneous emission of nearby nanostructured black phosphorus

Abstract

1. Introduction

2. Theoretical method

3. Spontaneous emission of a quantum emitter nearby BP sheet

4. Spontaneous emission of a quantum emitter nearby BPRAs

5. Conclusions

Funding

Disclosures

References

Cited By

Figures (6)

Equations (16)

Optics Express